The present disclosure relates to signal transmission and reception. There have been a number of approaches to the design of communication systems based on chaos, such as those suggested by Kocarev (1992), Belsky and Dmitriev (1993), Cuomo (1993), Pecora and Carrol (1993), and Dmitriev and Starkov (1997). These prior approaches have been focused on analog spread-spectrum types of systems and, hence, were inherently broadband. Moreover, there has been no attempt to restrict the state-space orbits of chaotic systems through symbolic constraints. Performance assessments have been done on these prior and other similar systems, but such systems have lacked symbolic dynamic controls or channel bandwidth controls.
The following are some definitions provided to enhance understanding of the descriptions that follow:
In geometry, linearity refers to Euclidean objects such as lines, planes, (flat) three-dimensional space, and the like. These objects appear the same no matter how they are examined. A nonlinear object, such as a sphere, for example, looks different for different scales. When viewed closely, it looks like a plane; and from afar, it looks like a point. In algebra, linearity is defined in terms of functions that have the properties f(x+y)=f(x)+f(y) and f(ax)=a f(x). Nonlinearity is defined as the negation of linearity. This means that the result f(x+y) may be out of proportion to the inputs x and/or y. Thus, nonlinear systems do not follow superposition themes.
A dynamical system has an associated abstract phase space or state space with coordinates that describe the dynamical state at any instant; and a. dynamical rule that specifies the immediate future trend of all state variables, given the present values of those state variables. Dynamical systems are “deterministic” if there is a unique consequent to every state; and “stochastic” or “random” if there is more than one consequent, typically chosen from some probability distribution. A dynamical system can be defined with respect to discrete or continuous time. The discrete case is defined by a map, z1=f(z0), which gives the state z1 resulting from the initial state z0 at the next discrete time value. The continuous case is defined by a “flow”, z(t)=φ(t)(z0)k, which gives the state at time t given that the state was z0 at time 0. A smooth flow can be differentiated with respect to (“w.r.t.”) time to give a differential equation, dz/dt=F(z). In this case, F(z) is called a vector field, which gives a vector pointing in the direction of the velocity at every point in a phase space.
A phase space or state space is the collection of possible states of a dynamical system. A state space can be finite (e.g., for the ideal coin toss, there are two states, heads and tails), countably infinite (e.g., where the state variables are integers), or uncountably infinite (e.g., where the state variables are real numbers). Implicit in the notion of state or phase space is that a particular state in phase space specifies the system completely. It is all one needs to know about the system to have complete knowledge of the immediate future.
Thus, the phase space of the planar pendulum is two-dimensional, consisting of the position or angle and velocity. Note that in a non-autonomous system where the map of the vector field depends explicitly on time (e.g., a model for plant growth that depends on solar flux), then, according to the definition of phase space, time must be included as a phase space coordinate since one must specify a specific time (e.g., 3 pm on Tuesday) to know the subsequent motion. Thus dz/dt=F (z,t) is a dynamical system on the phase space consisting of (z,t), with the addition of the new dynamic dt/dt=1. The path in phase space traced out by a solution of an initial value problem is called an orbit or trajectory of the dynamical system. If the state variables take real values in a continuum, the orbit of a continuous-time system is a curve; while the orbit of a discrete-time system is a sequence of points.
The notion of degrees of freedom as is used for Hamiltonian systems means one canonical conjugate pair: a configuration, q, and its conjugate momentum, p. Hamiltonian systems always have such pairs of variables and so the phase space is even-dimensional. In dissipative systems, the term phase space is often used differently to designate a single coordinate dimension of phase space.
A map is a function f on the phase space that gives the next state f(z) (i.e., the “image”) of the system given its current state z. A function must have a single value for each state, but there may be several attempt states that give rise to the same image. Maps that allow every state of the phase space to be accessed onto and which have precisely one pre-image for each state (i.e., a one-to-one correspondence) are invertible. If, in addition, the map and its inverse are continuous with respect to the phase space coordinate z, then it is called a homeomorphism. Iteration of a map means repeatedly applying the consequents of the previous application. Thus producing the sequence:Zn=f(zn−1)=f(f(Zn−2) . . . )=f(z0)where this sequence is the orbit or trajectory of the dynamical system with initial condition z0.
Every differential equation gives rise to a map. The time1 map advances the flow one unit of time. If the differential equation contains a term or terms periodic with time T, then the time T map in a system represents a Poincare section. This map is also called a stroboscopic map as it is effectively looking at the location in phase space with a stroboscope tuned to the period T. This is useful as it permits one to dispense with time as a phase space coordinate.
In autonomous systems (i.e., no time dependent terms in the equations), it may also be possible to define a Poincare section to reduce the phase space coordinates by one. Here, the Poincare section is defined not by a fixed time interval, but by successive times when an orbit crosses a fixed surface in the phase space. Maps arising out of stroboscopic sampling or Poincare sections of a flow are necessarily invertible because the flow has a unique solution through-any point in phase space. Thus, the solution is unique both forward and backward in time.
An attractor is simply a state into which a system settles, which implies that dissipation is needed. Thus, in the long term, a dissipative dynamical system may settle into an attractor. An attractor can also be defined as the phase space that has a neighborhood in which every point stays nearby and approaches the attractor as time goes to infinity. The neighborhood of points that eventually approach the attractor is the “basin of attraction”.
Chaos is defined as the effective unpredictable long term behavior arising in a deterministic dynamic system due to its sensitivity to initial conditions. It must be emphasized that a deterministic dynamical system is perfectly predictable given knowledge of its initial conditions, and is in practice always predictable in the short term. The key to long-term unpredictability is a property known as sensitivity to initial conditions. For a dynamical system to be chaotic, it must have a large set of initial conditions that are highly unstable. No matter how precisely one measures the initial conditions, a prediction of its subsequent motion eventually goes radically wrong.
Lyapunov exponents measure the rate at which nearby orbits converge or diverge. There are as many Lyapunov exponents as there are dimensions in the state space of the system, but the largest is usually the most important. Roughly speaking, the maximal Lyapunov exponent is the time constant λ in the expression for the distance between two nearby orbits. If λ is negative, the orbits converge in time and the dynamical system is insensitive to initial conditions. If λ is positive, then the distance between nearby orbits grows exponentially in time and the system becomes sensitive to initial conditions.
Lyapunov exponents can be computed in two ways. In one method, one chooses two nearby points and evolves them in time measuring growth rates of the distance between them. This method has the disadvantage that growth rate is not really a local effect as points separate. A better way to measure growth is to measure the growth rate of the tangent vectors to a given orbit. One defines
  λ  =            1      k        ⁢                  ∑                                      ⁢                          ⁢              ln        ⁢                                  ⁢                                                      f              ′                        ⁡                          (                              x                ⁡                                  (                  j                  )                                            )                                                    for j=0 to k−1. If λ is >0, it gives the average rate of divergence, or, if λ<0, it shows convergence.
The Minimum Phase Space dimension for Chaos is a slightly confusing topic, since the answer depends on the type of system considered. A flow or a system of differential equations is considered first. In this case, the Poincare-Bendixson theorem indicates that there is no chaos in one or two-dimensional phase space. Chaos is possible only in three-dimensional flows. If the flow is non-autonomous (i.e., dependent on time), then time becomes a phase space co-ordinate. Therefore, a system with two physical variables plus a time variable becomes three-dimensional and chaos is possible.
For maps, it is possible to have chaos in one dimension only if the map is not invertible. A prominent example would be a logistic map:x′=f(x)=rx(1−x)This equation is provably chaotic for r=4 and many other values. Note that for every point f(x)<½, the function has two pre-images and hence is not invertible. This concept is important as this method will be used to characterize various topologies of circuits used in realizing a system.
Higher order modulation systems such as M-ary phase shift keying (“PSK”) and M-ary quadrature amplitude modulation (“QAM”) require high levels of channel linearity in order to be successfully deployed. M-ary PSK and QAM systems are expensive to deploy due to the complexity of the system needed to make the architecture compliant with Federal Communications Commission (“FCC”) spectral templates. Systems with M-ary QAM or PSK architectures have a “set-top” box to decode the high-speed sub-carrier signals because normal receivers use FM demodulators for recovering the baseband information. In addition, M-ary systems suffer from power loss associated with higher levels of bandwidth compression brought about by the modulation scheme that employ multiple bits per symbol. The M-ary systems become too lossy for practical implementation beyond an upper limit.